Physical interpretation of intercore crosstalk in multicore fiber: effects of macrobend, structure fluctuation, and microbend

Tetsuya Hayashi; Takashi Sasaki; Eisuke Sasaoka; Kunimasa Saitoh; Masanori Koshiba

doi:10.1364/OE.21.005401

1. Introduction

Spatial division multiplexing using a multi-core fiber (MCF) is a strong candidate technology to overcome the capacity limit of single-core fiber transmission systems [1]. Inter-core crosstalk (XT) is one of the most important properties of uncoupled MCFs, and suppression of the XT has been actively studied [2–10]. Recently, Koshiba et al. derived a closed-form expression of the average power-coupling coefficient between cores in the MCF [11]. The closed-form expression is powerful, and easy to estimate the average crosstalk; however, it is difficult to interpret physical meaning of the expression intuitively.

In this paper, we derive another expression of the average power-coupling coefficient, whose physical meaning is easily interpreted. Based on the derived expression, we discuss how the structure fluctuation and macrobend can affect the crosstalk, and explain previously reported methods for crosstalk suppression. We also discuss how the microbending can affect the crosstalk in homogeneous and heterogeneous MCFs, based on the derived expression and previously reported measurement results.

2. Review and clarification of previous derivation of power-coupling coefficient

In this section, we review the derivation of the power-coupling coefficient between cores from the longitudinally perturbed coupled-mode equation. We also clarify an ambiguous point of the definition of power spectral density in the derivation.

Since the propagation constants of cores in the MCF are perturbed by bend, twist, structure fluctuation, and so on, the coupling between cores in the MCF can be described using the coupled power equation. The power-coupling coefficient can be derived from the coupled mode equation with longitudinally perturbed propagation constants. The coupled mode equation can be expressed as

\begin{matrix} \frac{d A_{n}}{d z} = - j κ_{n m} \exp [- j \int_{0}^{z} (β_{m} - β_{n}) d z] A_{m} \\ = - j κ_{n m} \exp [- j (β_{c, m} - β_{c, n}) z - j \int_{0}^{z} (β_{v, m} - β_{v, n}) d z] A_{m}, \end{matrix}

where A is the complex amplitude, κ_nm the mode coupling coefficient from Core n to Core m, β = 2πn_eff/λ the propagation constant, n_eff the effective refractive index, and λ the wavelength. Subscripts c and v of β represent constant and variable perturbed parts of β, respectively

Based on Eq. (1), in case of low crosstalk, the crosstalk in amplitude within the fiber segment [z₁, z₂] can be expressed as

Δ x_{n m} = \frac{Δ A_{n}}{A_{m}} \approx - j κ_{n m} \int_{z_{1}}^{z_{2}} \exp [- j (β_{c, m} - β_{c, n}) z] f (z) d z,

by using

f (z) \equiv \exp [- j \int_{0}^{z} [β_{v, m} (z^{'}) - β_{v, n} (z^{'})] d z^{'}] .

Accordingly, the average crosstalk increase in power within the segment [z₁, z₂] can be expressed as

\begin{matrix} Δ X_{n m} = 〈 {| Δ x_{n m} |}^{2} 〉 = 〈 Δ x_{n m} Δ x_{n m}^{*} 〉 \\ \approx κ_{n m}^{2} \int_{z_{1}}^{z_{2}} \int_{z_{1}}^{z_{2}} \exp [- j (β_{c, m} - β_{c, n}) (z - z^{'})] 〈 f (z) f^{*} (z^{'}) 〉 d z d z^{'} \\ \approx κ_{n m}^{2} \int_{z_{1}}^{z_{2}} \int_{z_{1} - z^{'}}^{z_{2} - z^{'}} \exp [j Δ β_{c, n m} ζ] 〈 f (z^{'} + ζ) f^{*} (z^{'}) 〉 d ζ d z^{'} \\ \approx κ_{n m}^{2} \int_{z_{1}}^{z_{2}} d z^{'} \int_{- \infty}^{\infty} R_{f f} (ζ) \exp (j Δ β_{c, n m} ζ) d ζ \\ \approx κ_{n m}^{2} Δ z \int_{- \infty}^{\infty} R_{f f} (ζ) \exp (j Δ β_{c, n m} ζ) d ζ, \end{matrix}

where R_ff is the autocorrelation function (ACF) of f(z), Δz is z₂ − z₁, and the correlation length l_c of R_ff is assumed to be adequately shorter than Δz.

R_{f f} (ζ)

can be understood as the correlation between the coupled and non-coupled lights that are propagated for the length of ζ after the coupling. For example, where ζ >> l_c, the coupled and non-coupled lights becomes incoherent even if the lights are very coherent. Based on the Wiener–Khinchin theorem, the power spectrum density (PSD) is the Fourier transform of the ACF:

S_{f f}^{(\tilde{ν})} (\tilde{ν}) = \int_{- \infty}^{\infty} R_{f f} (ζ) \exp (j 2 π \tilde{ν} ζ) d ζ,

where

\tilde{ν} = n_{eff} / λ = β / (2 π)

represents the wave number (or spatial frequency) in the medium—whereas the propagation constant β is the angular wave number. Note that

\tilde{ν}

, n_eff, and β have common subscripts, e.g.,

{\tilde{ν}}_{c} = n_{eff,c} / λ = β_{c} / (2 π)

. To describe the PSDs with respect to

\tilde{ν}

and β with common expressions, we would like to define the PSD with respect to β, whose total power is equivalent to Eq. (5). From the Parseval’s theorem, the average power of f(z), or expected value of |f(z)|², is equivalent to the integral of the PSD over whole

\tilde{ν}

, and the following equation holds between f(z) and the PSDs of f(z):

\int_{- \infty}^{\infty} S_{f f}^{(\tilde{ν})} (\tilde{ν}) d \tilde{ν} = \int_{- \infty}^{\infty} S_{f f}^{(\tilde{ν})} (\tilde{ν}) \frac{d \tilde{ν}}{d β} d β = E [{| f (z) |}^{2}] = 1,

where E[·] represents the expected value. Therefore, in this paper, the PSD

S_{f f}^{(β)} (β)

with the scale of the propagation constant β (the angular wave number in the medium) is defined as:

S_{f f}^{(β)} (β) \equiv S_{f f}^{(\tilde{ν})} (\tilde{ν}) \frac{d \tilde{ν}}{d β} = \frac{1}{2 π} S_{f f}^{(\tilde{ν})} (\tilde{ν}) = \frac{1}{2 π} \int_{- \infty}^{\infty} R_{f f} (ζ) \exp (j β ζ) d ζ .

From Eqs. (4)–(7), the power-coupling coefficient can be expressed as

h_{n m} = \frac{Δ X_{n m}}{Δ z} \approx κ_{n m}^{2} S_{f f}^{(\tilde{ν})} (\frac{Δ n_{eff,c, n m}}{λ}) = κ_{n m}^{2} [2 π S_{f f}^{(β)} (Δ β_{c, n m})] .

Figure 1 shows the schematics of perturbations on β, or how β_v can vary. As shown in Figs. 1(a) and 1(b), the bend and the structure fluctuation can induce a slight change in β_v in one core, which can occur ether in the single-core fiber or in the MCF. In the single-core fiber, by assuming proper R_ff or S_ff for the perturbations shown in Figs. 1(a) and 1(b), Eq. (8) is utilized for analyzing the power coupling between modes in the multi-mode fiber, microbend loss—power coupling from the core modes to the cladding modes, and so on. In the MCF, as shown in Fig. 1(c), the bend can induce relatively large β_v in a core when assuming another core as a reference of the propagation constant. Fini et al. [4] and Hayashi et al. [5] assumed that β_v in the MCF is induced by the macrobend and twist of the MCF as

β_{v, n} = β_{c, n} \frac{x_{n} \cos θ_{f} (z) - y_{n} \sin θ_{f} (z)}{R_{b} (z)} = β_{c, n} \frac{r_{n} \cos θ_{n} (z)}{R_{b} (z)},

and investigated the crosstalk characteristics of the MCFs. Here, (x_n, y_n) and (r_n, θ_n) are the local Cartesian and polar coordinates of Core n in a fiber cross-section, respectively, θ_n = 0 is the radial direction of the macrobend, θ_f the angle between the x-axis and the radial direction of the macrobend, and R_b the macrobend radius of the MCF—that is, the distance between the center of the macrobend and the origin of the local coordinates.

Fig. 1 Schematics of perturbations on the propagation constant. (a) a slight change of the propagation constant in a core due to bend, (b) a slight change of the propagation constant in a core due to structure fluctuation, and (c) a bend-induced change of the propagation constant in a core when assuming another core as a reference of the propagation constant.

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However, it is not easy to assume a proper β_v, R_ff, or S_ff that can include the perturbations of both the bend and the structure fluctuation. Therefore, by assuming that R_ff includes only the effect of structure fluctuation and does not include that of macrobend and twist, Koshiba et al. investigated the effects of correlation length l_c and of the shape of the ACF R_ff on the average crosstalk μ_X [10,11]. They investigated some types of R_ff, and found that the exponential ACF (EAF)

R_{f f} (ζ) = \exp (- | ζ | / l_{c})

is proper for estimating actual μ_X of the MCFs. The EAF have been introduced to microbending loss analysis [12]. Since the PSD of the EAF is the Lorentzian distribution, the power-coupling coefficient was obtained from Eq. (8) as [10,11]:

\begin{matrix} h_{n m} (z) = κ_{n m}^{2} \frac{1}{π} \frac{1 / (2 π l_{c})}{{1 / (2 π l_{c})}^{2} + {[Δ {n^{'}}_{eff,c, n m} (z) / λ]}^{2}} = κ_{n m}^{2} 2 π \frac{1}{π} \frac{1 / l_{c}}{1 / l_{c}^{2} + {[Δ {β^{'}}_{c, n m} (z)]}^{2}} \\ = κ_{n m}^{2} \frac{2 l_{c}}{1 + {[Δ {β^{'}}_{c, n m} (z) l_{c}]}^{2}}, \end{matrix}

where Δβ´_c,_nm is β´_c,_n − β´_c,_m, and β´_c is redefined β_c that includes the effects of macrobend and twist:

{β^{'}}_{c, n} = β_{c, n} (1 + \frac{x_{n} \cos θ_{f} (z) - y_{n} \sin θ_{f} (z)}{R_{b} (z)}) = β_{c, n} (1 + \frac{r_{n} \cos θ_{n} (z)}{R_{b} (z)}) .

Average crosstalk μ_X estimated using coupled-power equation with the power-coupling coefficient of Eq. (11) may be valid in cases where changes of R_b and θ are gradual enough compared to l_c, since Δβ´_c,_nm—which is variable and includes macrobend and twist— is substituted to Δβ_c,_nm—which is constant— in Eq. (8).

By assuming constant R_b and twist rate, Koshiba et al. also analytically derived an average power-coupling coefficient h̅, which is averaged over θ, as [11]:

{\bar{h}}_{n m} = κ_{n m}^{2} \sqrt{2} l_{c} [1 / \sqrt{a (b + \sqrt{a c})} + 1 / \sqrt{c (b + \sqrt{a c})}],

a = 1 + {(Δ β_{c, n m} l_{c} - B_{n m} l_{c} / R_{b})}^{2} \approx 1 + l_{c} {(Δ β_{c, n m} l_{c} - β_{c, n} D_{n m} l_{c} / R_{b})}^{2},

b = 1 + {(Δ β_{c, n m} l_{c})}^{2} - {(B_{n m} l_{c} / R_{b})}^{2} \approx 1 + {(Δ β_{c, n m} l_{c})}^{2} - {(β_{c, n} D_{n m} l_{c} / R_{b})}^{2},

c = 1 + {(Δ β_{c, n m} l_{c} + B_{n m} l_{c} / R_{b})}^{2} \approx 1 + {(Δ β_{c, n m} l_{c} + β_{c, n} D_{n m} l_{c} / R_{b})}^{2},

B_{n m} = \sqrt{{(β_{c, n} x_{n} - β_{c, m} x_{m})}^{2} + {(β_{c, n} y_{n} - β_{c, m} y_{m})}^{2}},

where B_nm can be approximated as β_c,_nD_nm if β_c,_m/β_c,_n ≈1, D_nm is the center-to-center distance between Core m and Core n. They also reported that Eqs. (13)–(17) agreed well with measurement results. However, it is difficult to interpret physical meaning of Eqs. (13)–(17) intuitively.

3. Derivation of an intuitive expression of average power-coupling coefficient

To understand the physical meaning of the average power-coupling coefficient, we derive another expression of the average power-coupling coefficient in this section. For simplicity, the center of Core m is taken as the origin of the local coordinate, and accordingly Δβ´_c,_nm can be written as

Δ {β^{'}}_{c, n m} (R_{b}, θ_{n m}) = Δ β_{c, n m} + Δ β_{b, n m} (R_{b}, θ_{n m}),

Δ β_{b, n m} (R_{b}, θ_{n m}) = Δ β_{b, n m}^{dev} (R_{b}) \cos θ_{n m},

Δ β_{b, n m}^{dev} (R_{b}) = β_{c, n} \frac{D_{n m}}{R_{b}},

where θ_nm represents the angle between the radial direction of the bend and a line segment from Core m to Core n, Δβ_b,_nm the difference of β variation between Core m and Core n from the macrobend, and

Δ β_{b, n m}^{dev}

the peak deviation of Δβ_b,_nm.

Let $p_{θ_{n m}} (θ_{n m})$ and $p_{R_{b}} (R_{b})$ be the probability density functions of θ_nm and of R_b, respectively, along the MCF; by assuming that $p_{θ_{n m}} (θ_{n m})$ and $p_{R_{b}} (R_{b})$ are statistically independent, the twist of the MCF is gradual enough, and average crosstalk is adequately low; the average crosstalk μ_X,nm from Core m to Core n can be expressed as

\begin{matrix} μ_{X, n m} (L) \approx \int_{0}^{L} h_{n m} (z) d z \approx L [\frac{1}{L} \int_{0}^{L} h_{n m} (z) d z] \approx L E [h_{n m}] \\ \approx L \int_{0}^{\infty} p_{R_{b}} (R_{b}) {\bar{h}}_{n m} (R_{b}) d R_{b}, \end{matrix}

where the average power-coupling coefficient is

{\bar{h}}_{n m} (R_{b}) = \int_{0}^{2 π} p_{θ_{n m}} (θ_{n m}) h_{n m} (R_{b}, θ_{n m}) d θ_{n m} .

By assuming that the twist of the MCF is random enough and the MCF is adequately long,

p_{θ_{n m}} (θ_{n m})

can be assumed to be constant ( = 1/(2π)) over all θ_nm; therefore, by substituting

Δ β = - Δ β_{b, n m} (R_{b}, θ_{n m}) = - Δ β_{b, n m}^{dev} (R_{b}) \cos θ_{n m}

and using Eq. (8) and

\sin (arc \cos x) = \sqrt{1 - x^{2}}

, Eq. (22) can be rewritten as

\begin{matrix} {\bar{h}}_{n m} (R_{b}) = \int_{0}^{2 π} \frac{1}{2 π} h_{n m} (R_{b}, θ_{n m}) d θ_{n m} = \int_{0}^{2 π} \frac{1}{2 π} κ_{n m}^{2} 2 π S_{f f}^{(β)} [Δ {β^{'}}_{c, n m} (R_{b}, θ_{n m})] d θ_{n m} \\ = 2 \int_{0}^{π} κ_{n m}^{2} S_{f f}^{(β)} [Δ β_{c, n m} + Δ β_{b, n m}^{dev} (R_{b}) \cos θ_{n m}] d θ_{n m} \\ = 2 π \int_{- Δ β_{b, n m}^{dev}}^{Δ β_{b, n m}^{dev}} \frac{κ_{n m}^{2}}{π \sqrt{{[Δ β_{b, n m}^{dev} (R_{b})]}^{2} - Δ β^{2}}} S_{f f}^{(β)} (Δ β_{c, n m} - Δ β) d (Δ β) . \end{matrix}

where S_ff is the Lorentzian distribution as shown in Eq. (11). By using the arcsine distribution:

p_{Δ β_{b}} (Δ β_{c, n m}) = {\begin{matrix} \frac{1}{π \sqrt{{(Δ β_{b, n m}^{dev})}^{2} - Δ β_{c, n m}^{2}}}, & | Δ β_{c, n m} | \leq Δ β_{b, n m}^{dev}, \\ 0, & otherwise, \end{matrix}

which is the probability distribution of Δβ_b, Eq. (23) can be rewritten as

{\bar{h}}_{n m} (Δ β_{c, n m}, R_{b}) = κ_{n m}^{2} 2 π {(p_{Δ β_{b}} * S_{f f}^{(β)})}_{Δ β} (Δ β_{c, n m}) = κ_{n m}^{2} {(p_{Δ {\tilde{ν}}_{b}} * S_{f f}^{(\tilde{ν})})}_{Δ \tilde{ν}} (Δ {\tilde{ν}}_{c, n m}),

where the expression of

{(f * g)}_{x}

denotes the convolution of f and g with respect to x, and the expression with respect to

\tilde{ν}

is also shown for comparison. If we consider the case where PSD S_ff in Eq. (8) includes both the effects of the structure fluctuation and the macrobend, the convolution term in Eq. (25) may be understood as the PSD S_ff in Eq. (8).

Particularly where |Δβ_c,_nm| and the bandwidth of $S_{f f}^{(β)}$ are adequately smaller than $Δ β_{b, n m}^{dev}$ , S_ff becomes a narrow delta-function-like distribution and the convolution contains only a gradually varying part of $p_{Δ β_{b}} (Δ β_{c, n m})$ ; therefore, Eq. (25) can be approximated as

\begin{matrix} {\bar{h}}_{n m} (Δ β_{c, n m}, R_{b}) \approx κ_{n m}^{2} [2 π p_{Δ β_{b}} (Δ β_{c, n m})] = κ_{n m}^{2} p_{Δ {\tilde{ν}}_{b}} (Δ {\tilde{ν}}_{c, n m}) \\ \approx κ_{n m}^{2} \frac{2}{\sqrt{{(\frac{β_{c, n} D_{n m}}{R_{b}})}^{2} - Δ β_{c, n m}^{2}}} = κ_{n m}^{2} \frac{λ}{π \sqrt{{(\frac{n_{eff,c, n} D_{n m}}{R_{b}})}^{2} - Δ n_{eff,c, n m}^{2}}}, \end{matrix}

which is also obtained from Eq. (8) by approximating the PSD

S_{f f}^{(β)}

as the probability distribution of Δβ_b—shown in Eq. (24)— with constant R_b, as shown in [13]. In case of homogeneous MCFs (Δβ_c_,nm = 0), Eq. (26) is reduced to

{\bar{h}}_{n m} (R_{b}) \approx κ_{n m}^{2} \frac{2 R_{b}}{β_{c, n} D_{n m}} = κ_{n m}^{2} \frac{λ R_{b}}{π n_{eff,c, n} D_{n m}},

which coincides with Eq. (19) in [9] and Eq. (25) in [11]. The difference between Eq. (26) and Eq. (27) is less than 0.1 dB when Δβ_c < 0.21β_cD/R_b; therefore, Eq. (27) may be also used for estimating the crosstalk of a bent heterogeneous MCF with small Δβ_c.

Figure 2 shows comparisons between h̅ calculated by using Eq. (25) and h̅ calculated by using Eqs. (13)–(17). Figures 2(a) and 2(b) show the PSDs normalized with respect to the Lorentzian S_ff and to the arcsine distribution $p_{Δ β_{b}}$ , respectively. The Lorentzian and arcsine distributions represent the spectra of the perturbations induced by the structure fluctuation and by the macrobend, respectively. Solid lines represent h̅ calculated by using Eq. (25) and dashed lines represent h̅ calculated by using Eqs. (13)–(17); however, the solid lines and the dashed lines are overlapped, and we can only see the solid lines. Accordingly, it was clearly confirmed that Eq. (25) is equivalent to the expression of h̅ with Eqs. (13)–(17), and it can be also said that the set of Eqs. (13)–(17) is a closed-form solution of the convolution of the Lorentzian and the arcsine distribution.

Fig. 2 Comparisons between h̅ calculated by using Eq. (25) and h̅ calculated by using Eqs. (13)–(17). (a) h̅ normalized with respect to the Lorentzian, (b) h̅ normalized with respect to the arcsine distribution. Solid lines: h̅ calculated by using Eq. (25), dashed lines: h̅ calculated by using Eqs. (13)–(17). The solid lines and the dashed lines are overlapped.

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4. Crosstalk suppression methods related to macrobend and structure fluctuation

Based on the above derivations, it can be understood that the crosstalk is proportional to the power of the mode-coupling coefficient and to the PSD of the perturbations. Of course, the suppression of the mode-coupling coefficient is important and various ways were proposed for confining power into cores such as high-index and small-diameter core structure [3,6], hole- or trench-assisted core structures [9,14,15], and photonic-crystal structures [16–18].

The PSD can be intuitively explained as the amount of the phase matching. Accordingly, how to suppress the PSD can be understood as how to suppress the phase matching. In this section, the methods for suppressing the phase matching are described.

The phase matching suppression methods can be categorized into some types according to how to utilize what kind of the perturbations. Here, three types of suppression methods are explained in the following subsections. A schematic example of ${\bar{h}}_{n m} (Δ β_{c, n m}, R_{b})$ in Eq. (25) shown in Fig. 3 will help with understanding, along with Fig. 2.

Fig. 3 A schematic example of the average power-coupling coefficient h̅, as a function of the propagation constant mismatch Δβ_c and the curvature 1/R_b, in case that twist of an MCF is gradual and random enough. (a) a 3-dimensional plot, (b) a contour map of log(h̅). Thick solid lines in (b) are the thresholds between the phase-matching region and the non-phase-matching region.

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4.1 Utilization of the propagation constant mismatch

One is the method utilizing the propagation constant mismatch Δβ_c to suppress the phase matching [2,4,19]. As shown as the non-phase-matching regions in Fig. 3, Δβ_c larger than $Δ β_{b}^{dev}$ can prevent the bend-induced phase-matching between dissimilar cores, and can suppress the crosstalk [4,5]. In other words, for suppressing the crosstalk, the bending radius of the MCF has to be managed to be adequately larger than the critical bending radius R_pk [5,11]:

R_{pk} = \frac{B_{n m}}{| Δ β_{c, n m} |} \approx D_{n m} \frac{β_{c, n}}{| Δ β_{c, n m} |} = D_{n m} \frac{n_{eff,c, n}}{| Δ n_{eff,c, n m} |},

that is, R_b where Eq. (26) can be infinite, or the maximal R_b where the phase matching due to the macrobend can occur even if there is no structure fluctuation. Some margin from R_pk is needed for avoiding the phase matching induced by the spectral broadening of S_ff, due to the structural fluctuations. In heterogeneous MCFs, it is preferred if the correlation length l_c of the structural fluctuation can be elongated, because the spectral broadening of S_ff can be narrowed and the PSD leakage into the non-phase-matching region can be suppressed, as shown in Figs. 2(b) and 3.

If most part of an MCF is deployed in gentle-bend conditions, a slight difference in propagation constants or effective indices may be enough for the phase matching suppression [4].

Since the typical winding radii of fiber spools are around 10 cm and R_pk less than 10 cm requires very large difference in core structure [5], most of crosstalk measurements and transmission experiments reported in various papers are considered to have been conducted in the phase-matching region. A hexagonal MCF with three kinds of cores reported in [20] is an exception, but it has a large difference in optical properties between cores so that R_pk can be smaller than the bobbin radius of 140 mm.

Recently, Saitoh et al. reported that up to two kinds of dissimilar step-index cores can be designed to achieve R_pk around 5 cm while achieving a similar A_eff of around 80 µm² at 1550 nm, and other good optical properties [21]. Tu et al. also reported that up to two kinds of dissimilar trench-assisted cores can be designed to achieve R_pk around 5 cm while achieving a similar A_eff of around 100 µm² [22].

4.2. Utilization of the bend-induced perturbation

The bend can also be utilized for the phase matching suppression [9]. As shown in Figs. 2(a) and 3, enlargement of the bend-induced perturbation—caused by the increase of the curvature or the decrease of the bending radius— can spread the PSD and suppress the crosstalk even in case of homogeneous MCFs (Δβ_c = 0). Identical core structure is rather desirable for suppressing the PSD. The PSD changes gradually with the bend radius, and there is no drastic PSD increase like that around R_pk in case of heterogeneous MCFs, since the PSD is suppressed in the phase-matching region. As shown in Eq. (27), the average crosstalk of a homogeneous MCF is proportional to the average bending radius, where |Δβ_c,_nm| and the bandwidth of $S_{f f}^{(β)}$ are adequately smaller than $Δ β_{b, n m}^{dev}$ . Therefore, if the average bending radius of the MCF is managed to be smaller than a certain value, or if the MCF is deployed in bend-challenged conditions, low crosstalk can be achieved with identical cores.

4.3. Utilization of the longitudinal structural fluctuation

As shown in Figs. 2(b) and 3(a), the power spectrum of the perturbations is broadened by the longitudinal structural fluctuations. If the R_ff due to the structure fluctuation has a very short correlation length l_c, the power spectrum spreads broadly over the propagation constant mismatch Δβ_c, and thus the PSD may be suppressed even in case of an unbent homogeneous MCF (Δβ_c = 0, 1/R_b = 0). A homogeneous MCF utilizing the longitudinal structural fluctuations was conceptually proposed by Takenaga et al. as “quasi-homogeneous MCF” in [3,6]. To the author’s knowledge, the crosstalk suppression by the structural fluctuation has not been actually observed yet, because the bend-induced perturbations are much larger than the fluctuation induced perturbations in the measurement conditions. However, the structural fluctuation may work when the MCF is cabled and installed in very-gently-bent conditions. For example, $l_{c} Δ β_{b, n m}^{dev} ≃ l_{c} β_{c} D / R_{b}$ of 1.3, 1/3, 1.0 × 10⁻¹, 1.0 × 10⁻², and 1.0 × 10⁻³ correspond to 1-dB, 5-dB, 10-dB, 20-dB, and 30-dB decreases in h̅ from Eq. (27) at Δβ_c = 0, respectively.

5. Applicability of the average power-coupling coefficient for estimating microbend-affected crosstalk

In Sections 2 and 3, S_ff only includes structure fluctuation, and $p_{Δ β_{b}} (Δ β_{c, n m})$ includes macrobend perturbation that gradually varies in longitudinal direction. Based on the assumption that the macrobend perturbation is gradual enough compared to l_c, we may redefine the S_ff as the PSD of high frequency perturbations other than the macrobend perturbation, and thus S_ff may include not only the effect of structure fluctuation but also the effect of microbend, in Eqs. (13)–(17) and Eq. (25). In this case, the increase of the microbend can be understood as the decrease of the correlation length l_c of S_ff. The microbend may induce β perturbations both within a core (Fig. 1(a)) and between cores (Fig. 1(c)); and may be induced by various ways such as winding on a sandpaper-coated bobbin, winding on a wire mesh bobbin, and actual cabling. Therefore, the shape of S_ff for the structure fluctuation and the microbend could possibly be different from the Lorentzian—that is, S_ff only for the structure fluctuation— and depend on how the microbend is induced. Thus, we need to investigate further details of the effect of the microbend on the crosstalk through experiment.

However, as a first step, we will evaluate the effect of the microbend on the crosstalk by assuming that S_ff for the structure fluctuation and the microbend, in this paper.

Figure 4 shows the dependences of the average power-coupling coefficient h̅ on microbend conditions for a heterogeneous MCF-A [23] and for a homogeneous MCF-B [24]. Fiber properties of the MCFs are shown in Table 1 . For MCF-A, h̅ from the center core to an outer core was obtained. For MCF-B, the average of h̅s between the three pairs of the neighboring cores was obtained. The values of h̅ were obtained from values of measured average crosstalk and fiber length by using coupled power equation. The microbend was applied by winding the MCFs on a 140-mm-radius bobbin with sandpaper (grade P240) at winding tension T. h̅ at T = 0 N was measured using a 140-mm-radius bobbin without the sandpaper. h̅ of the heterogeneous MCF was increased by the microbend, but that of the homogeneous MCF was varied only slightly.

Fig. 4 Dependences of the microbend on the average crosstalk (average power-coupling coefficient) for MCF-A (heterogeneous) and MCF-B (homogeneous), measured by wavelength averaging [26] with 100-m fiber [23,24] at λ = 1550 nm.

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Table 1. Characteristics of the Evaluated MCFs

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These results may be well explained as the shortening of l_c by using Eqs. (13)–(17) or Eq. (25). Figure 5 shows comparisons of the average power-coupling coefficients h̅s obtained from the measurement results and those obtained from Eqs. (13)–(17). Figure 5(a) shows the dependences of h̅ in MCF-A on R_b, and on whether the microbend is applied or not—that is, difference of l_c— at R_b = 140 mm. When the microbend is not applied, l_c can be estimated to be around 3 cm for this measurement. When the microbend is applied, l_c can be estimated to be around 1–4 mm. Figure 5(b) shows the dependences of h̅ in MCF-A and MCF-B on the propagation constant mismatch Δβ_c and on whether the microbend is applied or not, at R_b = 140 mm. We can see that Δβ_c between dissimilar cores in MCF-A is in the non-phase-matching region and h̅ between dissimilar cores is increased by the decreasing of l_c, or by the broadening of the bandwidth of S_ff. On the other hand, Δβ_c in MCF-B is designed to be zero and in the center of the phase-matching region; therefore, h̅ in MCF-B is hardly affected by the decreasing of l_c, at least if l_c is larger than 1 mm. Based on this evaluation, l_c was shortened from around 3 cm to around 1–4 mm by applying the microbend in these experiments. Though this shortening of l_c did not affect h̅ in MCF-B at R_b = 140 mm, the shortening of l_c may decrease h̅ in the (quasi-)homogeneous MCF if R_b is adequately large. Thus, we may consider that the microbend is possible to be utilized for suppressing the crosstalk in a very straight homogeneous MCF.

Fig. 5 Comparisons of the average power-coupling coefficient h̅s obtained from the measurements and from Eqs. (13)–(17). (a) The dependences of h̅ in MCF-A on the bending radius R_b and on the microbend. (b) The dependence of h̅ in MCF-A and MCF-B on the propagation constant mismatch Δβ_c and on the microbend at R_b = 140 mm. Closed-marks: h̅ measured without the microbend, open-marks: h̅ measured with the microbend, triangulars: h̅ measured by averaging the crosstalk by rewinding 2-m fiber 10 times [5], circles: h̅ measured by wavelength averaging [26] with 100-m fiber [23,24]. Solid lines: h̅ calculated at l_c = 3 cm, dashed lines: h̅ at l_c = 4 mm, dotted lines: h̅ at l_c = 3 mm, dashed-dotted lines: h̅ at l_c = 2 mm, dashed-two dotted lines: h̅ at l_c = 1 mm,

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We evaluated the effect of the microbend on the crosstalk by assuming S_ff for the structure fluctuation and the microbend as the Lorentzian. However, further investigation of the effect of the microbend on the crosstalk is necessary for elucidating the detailed characteristics of the effect, as we mentioned above.

6. Conclusions

We derived an intuitively interpretable expression of the average power-coupling coefficient as the convolution of the arcsine distribution—the spectrum of the perturbation induced by the macrobend— and the Lorentzian distribution—the spectrum of the perturbation induced by the structure fluctuation. The derived expression was confirmed to be equivalent to the closed-form expression derived in [11]. Based on the newly derived expression, we showed how the structure fluctuation and macrobend can affect the crosstalk, and organized previously reported methods for crosstalk suppression. We also discussed how the microbend can affect the crosstalk in homogeneous and heterogeneous MCFs. Though the average power-coupling coefficient affected by the microbend may be the convolution of the arcsine distribution and the spectrum of the high frequency perturbation, including the effects of the structure fluctuation and the microbend, we evaluated previously reported measurement results based on the derived expression—the convolution of the arcsine distribution and the Lorentzian distribution, as a first step. The crosstalk increase due to the microbend in the non-phase-matching region of the heterogeneous MCF and the crosstalk insensitivity to the microbend in the homogeneous MCF can be explained by the shortening of the correlation length of the high frequency perturbation. Further investigation on the effect of the microbend on the crosstalk will be reported in the future.

Appendix: Calculation of Eq. (25) for Figs. 2 and 3

Based on the following relationships of the Fourier transform:

J_{0} (Δ β_{b} ζ) \overset{Fourier transform}{\to} p_{Δ {\tilde{ν}}_{b}} (Δ {\tilde{ν}}_{c}) = 2 π p_{Δ β_{b}} (Δ β_{c}),

R_{f f} (ζ) \overset{Fourier transform}{\to} S_{f f}^{(\tilde{ν})} (Δ {\tilde{ν}}_{c}) = 2 π S_{f f}^{(β)} (Δ β_{c}),

f \cdot g \overset{Fourier transform}{\to} {(F * G)}_{Δ \tilde{ν}} = \frac{1}{2 π} {(F * G)}_{Δ β},

where J₀(x) is the Bessel function of the first kind of order zero, and functions F and G represents the Fourier transform of functions f and g respectively; we calculated Eq. (25) numerically by performing the fast Fourier transform (FFT) on the following relations:

J_{0} (Δ β_{b} ζ) R_{f f} (ζ) \overset{Fourier transform}{\to} {(p_{Δ {\tilde{ν}}_{b}} * S_{f f}^{(\tilde{ν})})}_{Δ {\tilde{ν}}_{b}} (Δ {\tilde{ν}}_{c}) = 2 π {(p_{Δ β_{b}} * S_{f f}^{(β)})}_{Δ β_{b}} (Δ β_{c}) .

We calculated adequately broad bandwidths of the PSD so that the aliasing noise caused by the FFT can be suppressed in the plotted ranges.

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	MCF-A (heterogeneous)	MCF-B (homogeneous)
Core arrangement	7 cores arranged on equilateral triangular lattice	3 cores arranged on equilateral triangular lattice
Core Δ	0.38% (all core)	0.32% (all core)
Core diameter	8.1 µm (the center core) 9.4 µm (outer cores)	11.2 µm (all core)
Core pitch	30 µm	38 µm
Mode-coupling coefficient in design	^*9.8 × 10⁻¹ at 1550 nm (between the center core and an outer core)	1.0 × 10⁻¹ at 1550 nm
Difference of n_eff between cores in design	Δ0.046% at 1550 nm (between the center core and an outer core)	Δ0%
Cladding diameter	125 µm	125 µm
Coating diameter	248 µm	244 µm

Physical interpretation of intercore crosstalk in multicore fiber: effects of macrobend, structure fluctuation, and microbend

Abstract

1. Introduction

2. Review and clarification of previous derivation of power-coupling coefficient

3. Derivation of an intuitive expression of average power-coupling coefficient

4. Crosstalk suppression methods related to macrobend and structure fluctuation

4.1 Utilization of the propagation constant mismatch

4.2. Utilization of the bend-induced perturbation

4.3. Utilization of the longitudinal structural fluctuation

5. Applicability of the average power-coupling coefficient for estimating microbend-affected crosstalk

6. Conclusions

Appendix: Calculation of Eq. (25) for Figs. 2 and 3

References and links

Cited By

Figures (5)

Tables (1)

Equations (32)

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